Question: $\int (5 x^4 +3 x^3 -2)\,dx=$ $+C$
We can use the sum rule and the constant multiple rule for indefinite integrals: $\begin{aligned} &\int [f(x)+g(x)]dx=\int f(x)\,dx+\int g(x)\,dx \\\\\\ &\int k\cdot f(x)= k\cdot\int f(x)\,dx \end{aligned}$ Using the sum and the constant multiple rules, we can rewrite our integral as follows: $\int (5 x^4 +3 x^3 -2)\,dx= 5\int x^4\,dx +3\int x^3\,dx -2\int 1\,dx$ Now we can find each indefinite integral using the reverse power rule: $\int x^n\,dx=\dfrac{x^{n+1}}{n+1}+C$ Note: we can only use the reverse power rule because $n \neq -1$. $\begin{aligned} &\phantom{=}\int (5 x^4 +3 x^3 -2)\,dx \\\\ &= 5\int x^4\,dx +3\int x^3\,dx -2\int 1\,dx \\\\ &=5 \dfrac{x^5}{5} +3\dfrac{x^4}{4} -2\dfrac{x^1}{1}+C \\\\ &=x^5 +\dfrac{3}{4} x^4 -2 x+C \end{aligned}$ In conclusion, $\int (5 x^4 +3 x^3 -2)\,dx=x^5 +\dfrac{3}{4} x^4 -2 x+C$